dimensions of well-being and their statistical measurements · 2016. 10. 13. · a quasi - newton...

Maurizio Vichi

Department of Statistical Sciences

Sapienza University of Rome

em: [email protected]

Dimensions of Well-Being and Their

Statistical Measurements: Statistical Composite Indicators to convey consistent policy messages

1

2016 Annual meeting of Community of Practice on

Composite Indicators and Scoreboards, 29 September 2016

A model-based Composite Indicator (CI) which is the result of the joint dimensional

reduction of the observed multivariate data.

The methodology has two aims:

- Indicator reduction: find a hierarchical simple structure model to identify a CI

- Units reduction: obtain the largest number of clusters with CI statistically different

Methodology: Clustering & Hierarchical Disjoint Non-negative Factor analysis

Properties of the CHDNFACHDNFA detects a General and some Specific Composite Indicators that best (MLE)

reconstruct the observed indicators (via a reflective model : data=CI model + error);

CHDNFA is scale equivariant, thus normalization of observed indicators does not effect the final

composite indicator;

CHDNFA identifies unique (latent) composite indicators which interpretation cannot be improved

by any orthogonal transformation;

CHDNFA produces reliable composite indicators by the best non-negative loadings;

CHDNFA defines Unidimensional Composite Indicators;

CHDNFA detects Composite Indicators with a robust ranking of individuals by means of

clustering. 2

Outline of the Presentation

3

Hierarchical simple structure model

to construct a Composite Indicator

4

Hierarchical simple structure model

General Composite Indicator

Specific Composite Indicators

Each factor

synthesises a set of

indicators Each factor

synthesises a set of

indicators or factors

Two level Hierarchical Simple Structure Model

5

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factor

variables

Observed

Indicators

Factor

Loadings

e1 e10e2 e3 e4 e5 e6 e7 e8 e9 e11 e12 e13 e14 e15

Variance

of error

Composite Indicator 1 Composite Indicator 2 Composite Indicator 3

F1General

Factor


Confirmatory approach: associations the number of specific factors, and association between

observed indicators and specific composite indicators (represented by arrows) are supposed known,

the level of correlation has to be estimated.

6

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factor

variables

Observed

Indicators

Factor

Loadings


Variance

of error


F1General

Factor


ESTIMATION: correlation between variables and factors, between general and specific factors

some associations may be not statistically significant (correlations are substantially null) and FIT POOR


0.7 0.60.5 0.70.5

0.20.7 0.3

0.70.30.8

0.2 0.2

0.80.2 0.7

7

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factor

variables

Observed

Indicators

Factor

Loadings


Variance

of error


F1General

Factor



0.7 0.60.5 0.70.5

0.20.7 0.3

0.60.40.7

0.2 0.2

0.80.2 0.7

At this point the researcher start to play with different models hypothesizing some changes that do not

have a theory behind.

8

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factor

variables

Observed

Indicators

Factor

Loadings


Variance

of error


F1General

Factor



0.7 0.60.5 0.70.5

0.30.7 0.4

0.60.40.7

0.30.4 0.8

0.2 0.7

The final model is one obtained by the researcher only “partially” sustained by a theory. The

modification is not ‘optimised’ and thus us the model selection becomes an “artisanal skill” of the

researcher

OUR PROPOSAL (1/3): Only the # of specific CIs is known

9

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factors

Observed

Indicators

Factor

Loadings


Variance

of error

Association between CIs and between CIS and Indicators are unknown

FgGeneral

Factor


? ? ?

?


10

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factors

Observed

Indicators

Factor

Loadings


Variance

of error

Association between CIs and observed Indicators are partially known

FgGeneral

Factor


? ? ?

?


OUR PROPOSAL (2/3) SOME FLEXIBILITY: also part of associations

are known, because these are sustained by a theory.

11

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Specific

Factors

Observed

Indicators

Factor

Loadings


Variance

of error

NON-NEGATIVE CORRELATIONS

FgGeneral

Factor


? ? ?

?


OUR PROPOSAL (3/3) Statistical Coherence of correlations.

Weights must be non-negative, unidimensionality and reliability of specific CIs

12

Clustering &

Hierarchical Disjoint Factor Analysisa model to identify the latent Hierarchical Composite Indicator and set of specific

Composite Indicators that best reconstruct the observed data and specify a ranking

of clusters

SOME METODOLOGICAL CONSIDERATIONS

13

Clustering &

Hierarchical Disjoint Factor Analysis

x x = Ay + ex, (y Specific factors) (1) y = cg + ey, (g General factor) (2)

Let include model (2) into model (1) the loading matrix A is

restricted to the product A=BV

Including (4) in HFA the HDFA model is defined

x x = BV(cg + ey) + ex = BVcg + BVey + ex. (3)

Let rewrite the model in matrix form

X = 𝐠𝐜′VB + Ex. (4)

Additionally the general factor scores g is partitioned into K disjoint clusters,

where matrix U is the membership matrix and 𝐠 is the centroid vector

X = 𝐔𝐠 𝐜′VB + Ex, (5)

with

Σx = BVc1

𝑛(𝐠 ′𝐔′𝐔𝐠 )𝐜′VB + x, (6)

where Σy = c

1

𝑛(𝐠 ′𝐔′𝐔𝐠 )c'+y. (7)

such that

V =[ vjh : vjh {0,1}] (binary) (8) V1H = 1J (row stochastic) (9)

U =[ ujk : uik {0,1}] (binary) (8)

U1K = 1n (row stochastic) (9)

B = diag(b1, …,bJ) with 𝑏𝑗2 > 0 (diagonal, non-null) (10)

V′BBV =diag(2

1.b ,…,2

.Hb ),with 01

22

.

J

j

jhh bb (orthogonal, non-empty) (11)

14Estimation of Clustering & HDFA

Minimization of the discrepancy functions w.r.t. B, V, U, 𝐘 and

Least-Squares Estimation

LSE(B,V, , U, 𝐘 ) = ||S - BV1

𝑛(𝐠 ′𝐔′𝐔𝐠 ))VB - x||

2 min (12)

B,V, , U, 𝐘

Maximum likelihood Estimation

MLE(B,V, , U, 𝐘 ) = J''n

trln'''n

ln

SΨBVgUUgBVSΨBVgUUgBV

1

')(1

||)(1 min (12)

B,V, , U, 𝐘

Generalised Least-Squares Estimation

GLSE(B,V, , U, 𝐘 ) = ||(S - BV1

𝑛(𝐠 ′𝐔′𝐔𝐠 )VB - x)S

-1/2||2 min (12)

B,V, , U, 𝐘

such that

V =[ vjh : vjh {0,1}] (binary) (13)

V1H = 1J (row stochastic) (14)

U =[ uik : vjk {0,1}] (binary) (15)

U1K = 1n (row stochastic) (16)

B = diag(b1, …,bJ) with 𝑏𝑗2 > 0 (diagonal, non-null) (17)

V′BBV =diag(2

1.b ,…,2

.Hb ),with 01

22

.

J

j

jhh bb (orthogonal, non-empty) (18)

A coordinated descendent algorithm has been developed this problem.

NOTE: This is a discrete and continuous problem that cannot be solved by

a quasi-Newton type algorithm

15

APPLICATION on WELL-BEINGOECD defines a Well-Being Index called Better Life index, considering 34 Countries and 24

indicators essential to identify well-being.

OECD, following Stiglitz et al. Committee (2009) identifies two dimensions

- Material Living Conditions (MLC)

- Quality of Life (QL)

OECD assumes that 8 observed variables define MLC and

the remain 16 variables the QL.

- Variables have been standardized. Min-max normalization can be used giving the same

results because the scale equivariance property.

OECD observes that some manifest variables are imperfect proxies of the concepts that one

would like to measure (MLC and QL) (Dolan, P., Peasgood, T. and White, M. (2008))

OECD leaves individual users to give subjective weighs to the variables

First a two-level hierarchy completely confirmatory model has been applied, requiring weights

to be non-negative. 8 indicators have weight zero (ex. Dwellings without basic facilities). This

means that they measure a negative component of the WB and for coherence must be

reversed. Then the model has been reapplied. Housing expenditure and Consultation on

rule-making have not significant correlation and therefore have been discarded. Then the

model has been reapplied with 22 indicators.

16Application: Better Life Index, OECD 201534 Countries, 22 indicators reflecting the pillars essential to identify well-being

Material Living Conditions

(Housing: (1) Dwellings without basic facilities; (2) Rooms per person); (Income: (3) Household net adjusted disposable income; (4) Household net

financial wealth); (Jobs: (5) Employment rate; (6) Job security; (7) Long-term unemployment rate; (8) Personal earnings);

Quality of Life (QL)

(Community: (9) Quality of social support network); (Education: (10) Educational attainment; (11) Student skills; (12) Years in education);

(Environment: (13) Air pollution; (14)Water quality); (Civic engagement: (15) Voter turnout); (Health: (16) Life expectancy; (17) Self-

reported health); (Life Satisfaction: (18) Life satisfaction); (Safety: (19) Assault rate; (20) Homicide rate); (Work-Life Balance: (21)

Employees working very long hours; (22)Time devoted to leisure and personal care)

F2 F1

1 2 3 4 5 14 15 16 17 18 2019 21 22

Specific

Factors

Factor

Loadings

e1e9e2 e3

e4 e5 e6 e7 e8 e10 e11 e20 e21 e22

FgGeneral

FactorWell-Being

6 7 1312111098

e12 e13 e14 e15 e16 e17 e18 e19

Material Living ConditionsCronbach’s alpha = 0.87Communality = 3.9463

Unidimensionality = 1.88

Quality of lifeCronbach’s alpha = 0.86Communality = 4.8719


0.610

0.803

0.946

0.747

0.630

0.2720.333

0.943 0.6850.621

0.6750.615

0.5630.781

0.2060.586

0.290 0.4380.633 0.668 0.670

0.539

0.8860.890

BIC = 6725.06

AIC = 6618.21

Disc.= 192.5

0.6270.353

0.1040.441

0.6020.925

0.8880.109 0.529

0.6130.543

0.6210.682

0.3890.957

0.6560.915

0.8070.598

0.5520.550

0.709

17

Can we obtain a better result?

- find a composite indicator that best reconstruct the 22

initial variables by using two dimensions?

- Can we found more than two dimensions?

Additional information: (11 dimensions),

3 define MLC; and 8 define QL

18Application: Better Life Index, OECD 201534 Countries, 22 indicators

Quality of Life (QL) )

(Housing: (1) Dwellings without basic facilities; (Jobs: (5) Employment rate; (6) Job insecurity; (7) Long-term unemployment rate;

(Community: (9) Quality of social support network); (Education: (10) Educational attainment; (11) Student skills; (12) Years in

education); (Environment: (13) Air pollution; (14) Water quality); (Health: (16) Life expectancy; (Life Satisfaction: (18) Life satisfaction;

(Civic engagement: (20) Homicide rate); (Work-Life Balance: (21) Employees working very long hours; (22) Time devoted to leisure

and personal care)

Material & Desired Living Conditions

(Housing: (2) Rooms per person); (Income: (3) Household net adjusted disposable income; (4) Household net financial wealth (Job:(8)

Personal earnings); (15) Voter turnout); (17) Self-reported health); (Safety: (19) Assault rate;

F1 F2

1 5 6 7 2220 21 2 3 4 158 17 19

Specific

Factors

Factor

Loadings

e1e12e5 e6

e7 e22e9 e10 e11e13 e16 e15 e17 e19

FgGeneral

FactorWell-Being

9 10 181614131211

e18 e20 e21e14 e2 e3 e4 e8

Quality of lifeCronbach’s alpha = 0.88

Communality = 5.2274


Material & (Desired) Living Conditions

Cronbach’s alpha = 0.88Communality = 3.5941

Unidimensionality =1.13

0.61

0.85

0.41

0.39

0.60

0.42

0.57

0.55

0.52 0.951 0.620.54 0.40

0.810.94

0.740.94 0.42 0.51

0.8830.880

BIC = 4741.67

AIC = 4634.83

Disc.= 134.163

0.5940.325

0.1520.501

0.0720.6820.807

0.817 0.5110.715 0.4940.845

0.865 0.282

0.38

0.8310.711

0.7090.771

0.0190.849

0.736

0.550.51


Quality of Life (QL) )

(Housing: (1) Dwellings without basic facilities; (Jobs: (5) Employment rate; (6) Job insecurity; (7) Long-term unemployment rate;

(Community: (9) Quality of social support network); (Education: (10) Educational attainment; (11) Student skills; (12) Years in

education); (Environment: (13) Air pollution; (14) Water quality); (Health: (16) Life expectancy; (Life Satisfaction: (18) Life satisfaction;

(Civic engagement: (20) Homicide rate); (Work-Life Balance: (21) Employees working very long hours; (22) Time devoted to leisure

and personal care)

Material & Desired Living Conditions

(Housing: (2) Rooms per person); (Income: (3) Household net adjusted disposable income; (4) Household net financial wealth (Job:(8)

Personal earnings); (15) Voter turnout); (17) Self-reported health); (Safety: (19) Assault rate;

F1 F2

1 5 6 7 2220 21 2 3 4 158 17 19

Specific

Factors

Factor

Loadings

e1e12e5 e6

e7 e22e9 e10 e11e13 e16 e15 e17 e19

FgGeneral

FactorWell-Being

9 10 181614131211

e18 e20 e21e14 e2 e3 e4 e8

Quality of lifeCronbach’s alpha = 0.88



Material & (Desired) Living Conditions

Cronbach’s alpha = 0.88Communality = 3.5941

Unidimensionality =1.13

0.61

0.85

0.41

0.39

0.60

0.42

0.57

0.55

0.52 0.951 0.620.54 0.40

0.810.94

0.740.94 0.42 0.51

0.8830.880

BIC = 4741.67

AIC = 4634.83

Disc.= 134.163

0.5940.325

0.1520.501

0.0720.6820.807

0.817 0.5110.715 0.4940.845

0.865 0.282

0.38

0.8310.711

0.7090.771

0.0190.849

0.736

0.550.51

20

0.000

500.000

1000.000

1500.000

2000.000

2500.000

0 5 10 15 20 25

pseudo F statistics

United States 1.99

Norway 1.71

Australia 1.57

Switzerland 1.56

Belgium 1.53

Germany 1.25

Denmark 1.08

New Zealand 1.04

Ireland 0.93

Japan 0.83

Finland 0.76

Netherlands 0.73

Luxembourg 0.61

Sweden 0.57

Austria 0.56

Italy 0.56

Iceland 0.42

Canada 0.41

United Kingdom 0.26

France 0.12

Greece -0.05

Korea -0.49

Slovenia -0.52

Spain -0.84

Poland -1.05

Israel -1.27

Mexico -1.3

Chile -1.49

Czech Republic -1.49

Slovak Republic -1.61

Hungary -1.66

Turkey -2.2

Portugal -2.21

Estonia -2.34

Ranking of clustersK=20 according to pF

Fg12

3

4

5

6

7

8

9

10

11

12

1314

15

16

1718

19

20


Material, Desired Living Conditions & Security

Material ,desired conditions (2) Rooms per person; (3) Household net adjusted disposable income; (4) Household net financial wealth; (8)

Personal earnings);

(15) Voter turnout); (16) (18) Life satisfaction); (22)Time devoted to leisure and personal care

Job security (6) Job insecurity; (7) Long-term unemployment rate

Quality of Life (QL) (Education, Society, Habitat)

Quality of Safety and Education: (10) Educational attainment; (11) Student skills; (19) Assault rate; (20) Homicide rate;

Quality of Society (9) Quality of social support network); (14)Water quality); (5) Employment rate; (12) Years in education;

Quality of the habitat (1) Dwellings without basic facilities (13) Air pollution; (17) Self-reported health; (21) Employees working very long hours;

F1 F3

2 3 4 8 15 19 20 5 9 12 1314 17 21

e2e3 e4

e8 e15 e16 e18 e22e6 e7 e13 e17 e21

FgWell-Being

16 18 11107622

e10 e11 e19 e20 e5 e9 e12 e14

Material, Desired Living Conditions & Security Quality of life (Education, Society, Habitat)

0.806

0.923

0.718

0.967

0.963

0.6940.708

0.3490.8591 0.757 0.924

0.7590.840

0.5630.545

0.9970.941

0.5800.703

0.924

0.8215

BIC = 4007.78

AIC = 3791.04

Disc.= 104.17

0.3490.146

0.4830.441

0.6020.925

0.8880.109 0.261 0.567 0.425

0.1440.423

0.2920.682

0.7020.004

0.6620.763

0.504

0.6570.8591

F5 F2 F4

e1

1

Cronbach’s alpha = 0.88


Unidim=1.024

0.7697 0.3451



Unid.im.=0.261



Unidim.= 0.619



Unidim.=0.770



Unidim.=0.795

0.6489

0.486

0.1140.261

Job security

Material & desired

living conditionsQuality of Education

& safety, Quality of Society

Quality of Habitat

22

CONCLUSIONCI to convey consistent policy messages

Properties of the CHDNFA

CHDNFA detects a General and some Specific Composite Indicators that best

(MLE) reconstruct the observed indicators (via a reflective model);

- CHDNFA is scale equivariant, thus allowing any scaling of indicators necessary

to normalise the observed indicators with different units of measurements;

- CHDNFA identifies unique (latent) composite indicators which interpretation

cannot be improved by any orthogonal transformation;

- CHDNFA produces reliable composite indicators by the best non-negative

loadings;

- CHDNFA, with the correct model selection, defines Unidimensional Composite

Indicators;

- CHDNFA detects Composite Indicators with a robust ranking of individuals by

means of clustering.

23

Thank you for your

attention!

24

(a) n=100,J=10, d=0.33 Low error – blocks are well visible

(b) n=100,J=10, d=1.00 Low error –blocks are well visible

(c) n=100, J=10, d=1.33 Medium error –blocks are visible

(e) n=100, J=10, d=1.66

Medium error –blocks are visible

(f) n=100, J=10, d=2.00

High error –blocks are not visible

(g) n=100,J=10, d=2.33

High error –blocks are not visible

(h) n=500, J=50, d=1.66 Low error

(i) n=500, J=50, d=2.00 Low error

(l) n=500, J=50, d=2.66 Medium error

(m) n=500,J=50, d=3.00

Medium error

(p) n=500,J=50, d=3.66

High error-blocks are not visible

(q) n=500,J=50, d=4.66

High error- blocks are not visible

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10 -0.5

0

0.5

1

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10-0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50 -0.5

0

0.5

1

SIMULATION STUDY

n=100, J=10, H=3

Three levels of error low, medium high

n=500, J=50 H=3

Three levels of error low, medium high

y ∼ NH(0, I) and

e ∼ NJ(0, d), (d = 0.1, 1, 2),

with j∼U(0, 1),

B=diag(b1,…,bJ)

with bj = 0.7sign(a) + 0.1a , with a ∼ N(0, 1),

V =[v1, ….,vJ]'

with vJ ∼ Multinomial( H: ph=1/H, h=1,…,H),

25Table 1: simulated data sets with n=100, J=10, H=3 and different level of error.

Error low Error medium Error high

d=0.33 d=0.66 d=1 d=1.33 d=1.66 d=2.00 d=2.33

ARI 1.000 1.000 0.997 0.973 0.934 0.809 0.640

GFI

0.923

0.919

0.930

0.921

0.921

0.923

0.924

AGFI 0.864 0.857 0.876 0.860 0.861 0.863 0.865

RMSEA 0.156 0.162 0.148 0.161 0.161 0.159 0.156

RMSR 0.0012 0.0034 0.0046 0.0055 0.0058 0.0060 0.006

BIC

-78.00

706.23

984.12

1149.68

1223.60

1268.50

1287.20

AIC -140.52 643.71 940.18 1087.15 1161.07 1205.98 1224.67

BICH1/BICH0 157 144 120 81 76 73 72

AICH1/AICH0 161 140 116 85 80 77 75

Table 2: simulated data sets with n=500, J=50, H=3 and different level of error.

Error low Error medium Error high

d=1.66 d=2.00 d=2.66 d=3.00 d=3.66 d=4.00 d=4.66

ARI 1.000 1.000 0.999 0.994 0.978 0.958 0.878

GFI

0.897

0.897

0.896

0.896

0.895

0.897

0.897

AGFI 0.883 0.884 0.883 0.882 0.882 0.884 0.884

RMSEA 0.069 0.069 0.069 0.069 0.069 0.069 0.069

RMSR 0.001 0.002 0.002 0.002 0.002 0.002 0.002

BIC

11480.53

21096.27

23385.81

26073.98

26745.66

27751.42

27957.30

AIC 10873.63 20489.36 22778.91 25467.08 26138.76 27144.52 27350.40

BICH1/BICH0 7.52 3.32 2.99 2.67 2.60 2.51 2.49

AICH1/AICH0 6.78 3.39 3.04 2.71 2.64 2.54 2.52

26

Cross-LoadingsThe fit of the SSM may be poor: for uncorrect choice of the number of factors

for the presence of cross-loadings

PROCEDURE: FIRST ESTIMATE the best SSM

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Observed

Indicators

Factor

Loadings


Variance

of error


Factors

latent

variables

Cross-Loadings- IDENTIFY the Cross-Loanding that most reduce BIC

27

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Observed

Indicators

Factor

Loadings


Variance

of error


Factors

latent

variables

Cross-LoadingsContinue to IDENTIFY Cross-Loadings that reduce BIC

28

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Observed

Indicators

Factor

Loadings


Variance

of error


Factors

latent

variables

29

Cross-LoadingsSTOP

29

F1 F2 F3

1 2 3 4 5 6 7 8 9 10 1211 13 14 15

Observed

Indicators

Factor

Loadings


Variance

of error

0

500

1000

1500

0 1 2 3 4 5

BIC

# inserted loadings

BIC of the SSM with cross-loadings

30

Cross-LoadingsExample: 20 x 20 correlation matrix

3 blocks generated according DFA + cross-loadings a10,1 , a1,3

Estimation:

first estimate the best Simple Structure Model

[3 blocks (V1-V4), (V5-13), (V14-V20)]

BIC = 1366, AIC = 785

First cross-loading estimated a10,1 =0.57;

BIC = 925, AIC = 785

Second cross-loading estimated a1,3 =0.52;

BIC = 585, AIC = 445

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

200

400

600

800

1000

1200

1400

1600

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

BIC

# inserted loadings

BIC of the SSM with cross-loadings

31

Recall that the discrepancy function D(B, 𝐕 , ) is minimized with respect to Bh=diag(bh) by

𝒃ℎ = 𝐱ℎ

−1

2u1h (1ℎ – 1)1

2 h=1,…,H.

where 1h is the largest eigenvalue and 𝐮1ℎ is the corresponding eigenvector of the variance-covariance

matrix 𝐱ℎ

−1

2 𝐒ℎ 𝐱ℎ

−1

2 corresponding to variables identified by v.h, that corresponds to h-th column of V.

Values 1h and 𝐮1ℎ minimize || 𝐱ℎ

−1

2 𝐒ℎ 𝐱ℎ

−1

2 − 1ℎ𝐮1ℎ𝐮1ℎ′ ||

2, or equivalently

|| 𝐗ℎ 𝐱ℎ

−1

2 − 1ℎ𝐲ℎ𝐮1ℎ′ ||

2, (52)

where Xh is the centered data matrix formed by variables identified by v.h and yh is the factor score vector.

ariables.

Hierarchical Disjoint Non-Negative Factor Analysis

The General factor frequently corresponds to a Composite indicator where each subset

of variable is consistent and reliable, thus the loadings must be positive

32

|| 𝐗ℎ 𝐱ℎ

−1

2 − 1ℎ𝐲ℎ𝐮1ℎ′ ||

2, (52)

can be solved by an ALS algorithm that alternates two regression problems.

Given 𝐮 1ℎ compute yh by

yh = 𝐗ℎ 𝐱ℎ

−1

2𝐮 1ℎ (𝐮 ℎ′ 𝐮 1ℎ )−1. (53)

Given 𝐲 ℎ compute 𝐮1ℎby

𝐮1ℎ = ℎ

−1

2𝐗ℎ′ 𝐲 ℎ(𝐲 ℎ

′ 𝐲 ℎ)−1. (54)

At each reiteration of the two steps (53) and (54), the loss function (52) decreases or at least does not

increase. The algorithm stops when function (52) decreases less than a positive arbitrary constant.

Now vector 𝐮𝟏𝒉 must be non-negative.

The solution can be found by the Non-Negative LS Algorithm (Lawson and Hanson, 1974).

This is an active set algorithm, where the H inequality constraints are active if 𝐮1ℎ′ are negative (or

zero) when estimated unconstrained, otherwise constraints are passive.

The non-negative solution of (52) with respect to 𝐮1ℎ is the unconstrained least squares solution

using only the variables of the passive set, setting the regression coefficients of the active set to

zero. Therefore

𝐮1ℎ = 𝐱ℎ

−1

2𝐗ℎ+′ 𝐲 ℎ(𝐲 ℎ

′ 𝐲 ℎ)−1

0 otherwise . (55)

where 𝐗ℎ+ is the set of passive variables.

SOLUTION OF a REGRESSION PROBLEM

dimensions of well-being and their statistical measurements · 2016. 10. 13. · a quasi - newton...

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